1. ## Partial Derivatives

If f(x,y,z) = 0 and g(x,y,z) = 0, show that these three fractions are equal:
$\displaystyle \frac{dx}{\frac{\partial(f,g)}{\partia l(y,z)}}$
$\displaystyle \frac{dy}{\frac{\partial(f,g)}{\partial(z,x)}}$
$\displaystyle \frac{dz}{\frac{\partial(f,g)}{\partia l(x,y)}}$

I know dF and dG both equal zero, but setting them both equal to zero and using Jacobians does not get me to the final answer. I don't know what else to try. Any help would be greatly appreciated!

I do not know how to get rid of those 68.. numbers.. btw

If f(x,y,z) = 0 and g(x,y,z) = 0, show that these three fractions are equal:
$\displaystyle \frac{dx}{\frac{\partial(f,g)}{\partial(y,z)}}$

$\displaystyle \frac{dy}{\frac{\partial(f,g)}{\partial(z,x)}}$

$\displaystyle \frac{dz}{\frac{\partial(f,g)}{\partial(x,y)}}$

I know dF and dG both equal zero, but setting them both equal to zero and using Jacobians does not get me to the final answer. I don't know what else to try. Any help would be greatly appreciated!

Here is a better version!

3. Originally Posted by cuteangel

If f(x,y,z) = 0 and g(x,y,z) = 0, show that these three fractions are equal:
$\displaystyle \frac{dx}{\frac{\partial(f,g)}{\partial(y,z)}}$

$\displaystyle \frac{dy}{\frac{\partial(f,g)}{\partial(z,x)}}$

$\displaystyle \frac{dz}{\frac{\partial(f,g)}{\partial(x,y)}}$
Okay, I took the cross product of the gradient of f and the gradient of g and found the partial derivative jacobians in the denominator. This vector will be parallel to the intersection of the tangent planes if the curve C is given as the intersection of f(x,y,z)=0 and g(x,y,z)=0. However, I suppose to find the equation I am after I need to take the cross product of these partial derivatives with (dx, dy, dz). But I do not know why or how this would apply. Any help would be greatly appreciated!